Graphical Representations Between Two Quantitative Variables

Two measurements on each individual

Until now you have graphed one quantitative variable at a time. Here every individual comes with two quantitative measurements: each car has an age AND a price. That is bivariate quantitative data. Each individual contributes one ordered pair $(x,y)$, and both numbers have to come from the same individual, whether the individuals are a sample or a whole population. Pairing one car's age with another car's price would be meaningless.

A scatterplot shows bivariate data with one dot per individual. The explanatory variable goes on the x-axis; it is the variable whose values you use to explain or predict the other one. The response variable goes on the y-axis.

Sometimes either variable could play either role. Height and arm span are a good example. Then the question you are asking decides: if you want to predict arm span from height, height is explanatory.

Four things to describe

A complete description of a scatterplot covers four features.

• Form. Do the points follow a straight-line path (linear) or a curve (non-linear)?
• Direction. In a positive association, as the explanatory variable increases, the response tends to increase. In a negative association, the response tends to decrease.
• Strength. How closely do the points follow the pattern? Describe it as strong, moderate or weak.
• Unusual features. Are there clusters, or individual points that do not fit the general pattern?

For the sedans, the association is linear, negative and strong, with no unusual features. In context: older cars of this model tend to have lower asking prices, and prices fall at a fairly steady rate as age goes up.

The words tend to are doing real work there. The 3-year-old car, at 24.2 thousand dollars, is priced higher than both 2-year-olds. A negative association describes the overall pattern, not a rule that every car obeys.

If the points form a shapeless cloud, there is no association, and so no form or direction to describe.

Strong is not the same as linear

Form trips up more students than the other three features. A student practices typing for four weeks and checks her speed every other day. Her speed climbs from 18 to 48 words per minute in the first 10 days, then adds only 5 more words per minute over the last 10. Points that rise steeply and then flatten out are non-linear, even when they follow their pattern very closely.

Clusters and stray points

Unusual features are often the most interesting part of a plot. A park ranger times 20 eruptions of a geyser, along with the wait until the next eruption. The plot does not show one cloud of points. It shows two.

Overall the association is positive and strong: longer eruptions tend to be followed by longer waits. A description that stopped there would hide the most important thing about this geyser, which seems to have two kinds of eruptions. Report the clusters, with their values.